On the Principles of Arithmetic and On the Principles of Music

5XXII. On the types of greater and lesser quantity.
XXIII. On the multiple, its types, and their origins.
XXIIII. On the superparticular original: superparticulari. A ratio where the larger number contains the smaller number once, plus one single part of it (e.g., 3:2, where 3 contains 2 plus half of 2)., its types, and their origins.
XXV. On a certain useful property relating to the understanding of superparticulars.
XXVI. A diagram through which it is taught that multiplicity is more ancient than the other types of inequality.
10XXVII. The reasoning and explanation of the arranged table.
XXVIII. On the third type of inequality, which is called the superpartient original: superpartiens. A ratio where the larger number contains the smaller number once, plus more than one part of it (e.g., 5:3, where 5 contains 3 plus two-thirds of 3)., and on its types and their origins.
XXVIIII. On the multiple superparticular.
XXX. On finding examples of these in the table above.
15XXXI. On the multiple superpartient.
XXXII. A demonstration of how every inequality proceeds from equality.

Here begins the first book.

20### Preface, in which the division of mathematics is discussed.

I. Among all the men of ancient authority who, with Pythagoras as their leader, flourished through a purer use of the mind's reason, it is clearly established that no one reaches the summit of perfection in the disciplines of philosophy unless they track the nobility of such wisdom through a certain "fourfold path" Quadrivium: the four mathematical arts of arithmetic, music, geometry, and astronomy., which will not escape the skill of one who looks at it correctly. For wisdom is the knowledge of things which exist, and which have obtained an immutable substance 25 as their lot—

2 "of them" [masculine/neuter] in manuscripts a, c, r, s. "of them" [feminine] in l. || "types" in r. 4 "of them" in b, c, r, s. 6 "kinship" in a, b, c; "knowledge" (with a dot placed under the 'i' and 'a' written above) in s. 8 "inequalities" in a. 10 "digested" in c, d, r. 11 "which" original: quae omitted in b. 12 "types" omitted in b. || "of them" in s. 14 "of the superior" in f, "in the superior" in r. 18 "The prologue of Boethius ends" in b. || "of the first book" omitted in c, r. || Heading omitted in d, f, l, s. 19 "Here begins the first book" omitted in b, f, r, d, l. 20 "Preface... mathematics" omitted in a, d, l. || "in which is the division" in c. 26 "skill of one who looks" in a, b, c, d, f, l.

Vocabulary: Boethius, Arithmetic, Pythagoras, Philosophy, Quadrivium, Wisdom, Inequality, Multiplicity